Colin leslie dean argues Gödel is a complete failure as he ends in utter meaninglessness Gödel’s incompleteness theorem ends in absurdity or meaninglessness, This is a case study demonstrating that all views end in meaninglessness. What do you think
http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdfGÖDEL’S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS GÖDEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS
CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS
By
COLIN LESLIE DEAN
B.SC, B.A, B.LITT (HONS), M.A, B,LITT (HONS), M.A,
M.A (PSYCHOANALYTIC STUDIES), MASTER OF PSYCHOANALYTIC STUDIES, GRAD CERT (LITERARY STUDIES)
GÖDEL’S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS GÖDEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS
CASE STUDY IN THE MEANINGLESSNESS OF ALL VIEWS
By
COLIN LESLIE DEAN
B.SC, B.A, B.LITT (HONS), M.A, B,LITT (HONS), M.A,
M.A (PSYCHOANALYTIC STUDIES), MASTER OF PSYCHOANALYTIC STUDIES, GRAD CERT (LITERARY STUDIES)
GAMAHUCHER PRESS WEST GEELONG, VICTORIA AUSTRALIA
2007
A case study in the view that all views end in meaninglessness. As an example of this is Gödel’s incompleteness theorem. Gödel is a complete failure as he ends in utter meaninglessness
What Gödel proved was not the incompleteness theorem but that mathematics was self contradictory. But he proved this with flawed and invalid axioms- axioms that either lead to paradox or ended in paradox –thus showing that Godel’s proof is based upon a misguided system of axioms and that it is invalid as its axioms are invalid.
Godel states “the most extensive formal systems constructed up to the present time are the systems of Principia Mathematica (PM) on the one hand and on the other hand the Zermel-Fraenkel axiom system of set theory … it is reasonable therefore to make the conjecture that these axioms and rules of inference are also sufficient to decide all mathematical questions which can be formally expressed in the given axioms. In what follows it will be shown that this is not the case but rather that in both of the cited systems there exist relatively simple problems of the theory of ordinary numbers which cannot be decided on the basis of the axioms” (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,pp.5-6)
All that he proved was in terms of PM and Zermelo axioms-there are other axiom systems -so his proof has no bearing outside that system he used Russell rejected some axioms he used as they led to paradox. All that Gödel proved was the lair paradox -which Russell said would happen
Gödel used impedicative definitions- Russell rejected these as they lead to paradox (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.63)
Gödel used the axiom of reducibility -Russell abandoned this as it lead to paradox (K. Godel, op.cit, p.5)
Gödel used the axiom of choice mathematicians still hotly debate its validity- this axiom leads to the Branch-Tarski and Hausdorff paradoxes (K.Godel, op.cit, p.5)
Gödel used Zermelo axiom system but this system has the skolem paradox which reduces it to meaninglessness or self contradiction
Godel proved that mathematics was inconsistent
from Nagel -"Gödel" Routeldeg & Kegan, 1978, p 85-86
Gödel also showed that G is demonstrable if and only if it’s formal negation ~G is demonstrable. However if a formula and its own negation are both formally demonstrable the mathematical calculus is not consistent (this is where he adopts the watered down version noted by bunch) accordingly if (just assumed to make math’s consistent) the calculus is consistent neither G nor ~G is formally derivable from the axioms of mathematics. Therefore if mathematics is consistent G is a formally undecidable formula Gödel then proved that though G is not formally demonstrable it nevertheless is a true mathematical formula
From Bunch
"Mathematical fallacies and paradoxes” Dover 1982" p .151
Gödel proved
~P(x,y) & Q)g,y)
in other words ~P(x,y) & Q)g,y) is a mathematical version of the liar paradox. It is a statement X that says X is not provable. Therefore if X is provable it is not provable a contradiction. If on the other hand X is not provable then its situation is more complicated. If X says it is not provable and it really is not provable then X is true but not provable Rather than accept a self-contradiction mathematicians settle for the second choice
Thus Godel by using invalid axioms i.e. those that lead to paradox or end in paradox only succeeded in getting the inevitable paradox that his axioms ordained him to get. In other words he could have only ended in paradox for this is what his axioms determined him to get. Thus his proof is a complete failure as his proof. that mathematics is inconsistent was the only result that he could have logically arrived at since this result is what his axioms logically would lead him to; because these axioms lead to or end in paradox themselves. All he succeeded in getting was a paradoxical result as Russell new would happen if those axioms where used. Godel by using those axioms could only arrived at a paradoxical result
Gödel used the Zermelo axiomatic system but this system end in meaninglessness. There is the Skolem paradox which collapses axiomatic theory into meaningless
Bunch notes op cit p.167
“no one has any idea of how to re-construct axiomatic set theory so that this paradox does not occur”
To give detail
Godel uses the axiom of reducibility and axiom of choice
Quote
http://www.mrob.com/pub/math/goedel.htm“A. Whitehead and B. Russell, Principia Mathematica, 2nd edition, Cambridge 1925. In particular, we also reckon among the axioms of PM the axiom of infinity (in the form: there exist denumerably many individuals), and the axioms of reducibility and of choice (for all types)” ((K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.5)
AXIOM OF REDUCIBILITY
(1) Godel uses the axiom of reducibility axiom 1V of his system is the axiom of reducibility “As Godel says “this axiom represents the axiom of reducibility (comprehension axiom of set theory)” (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,p.12-13.
( 2) “As a corollary, the axiom of reducibility was banished as irrelevant to mathematics ... The axiom has been regarded as re-instating the semantic paradoxes” -
http://mind.oxfordjournals.org/cgi/reprint/107/428/823.pdf 2)“does this mean the paradoxes are reinstated. The answer seems to be yes and no” -
http://fds.oup.com/www.oup.co.uk/pdf/0-19-825075-4.pdf )
AXIOM OF CHOICE
Godel states he uses the axiom of choice “this allows us to deduce that even with the aid of the axiom of choice (for all types) … not all sentences are decidable…” (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965. p.28.)
(“The Axiom of Choice (AC) was formulated about a century ago, and it was controversial for a few of decades after that; it may be considered the last great controversy of mathematics…. A few pure mathematicians and many applied mathematicians (including, e.g., some mathematical physicists) are uncomfortable with the Axiom of Choice. Although AC simplifies some parts of mathematics, it also yields some results that are unrelated to, or perhaps even contrary to, everyday "ordinary" experience; it implies the existence of some rather bizarre, counterintuitive objects. Perhaps the most bizarre is the Banach-Tarski Paradox “– http://www.math.vanderbilt.edu/~schectex/ccc/choice.html)
ZERMELO AXIOM SYSTEM
Godel specifies that he uses the Zermelo axiom system- (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,p.28.)
quote
http://www.mrob.com/pub/math/goedel.html "In the proof of Proposition VI the only properties of the system P employed were the following:
1. The class of axioms and the rules of inference (i.e. the relation "immediate consequence of") are recursively definable (as soon as the basic signs are replaced in any fashion by natural numbers).
2. Every recursive relation is definable in the system P (in the sense of Proposition V).
Hence in every formal system that satisfies assumptions 1 and 2 and is ω-consistent, undecidable propositions exist of the form (x) F(x), where F is a recursively defined property of natural numbers, and so too in every extension of such
[191]a system made by adding a recursively definable ω-consistent class of axioms. As can be easily confirmed, the systems which satisfy assumptions 1 and 2 include the Zermelo-Fraenkel and the v. Neumann axiom systems of set theory,47"
IMPREDICATIVE DEFINITIONS
Godel used impredicative definitions
Quote from Godel
“ The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... We saw that we can construct propositions which make statements about themselves,… ((K Godel , On undecidable propositions of formal mathematical systems in The undecidable , M, Davis, Raven Press, 1965, p.63 of this work Dvis notes, “it covers ground quite similar to that covered in Godels orgiinal 1931 paper on undecidability,” p.39.)
Godel used Peanos axioms but these axioms are impredicative and thus according to Russell Poincaré and others must be avoided as they lead to paradox.
quote
http://en.wikipedia.org/wiki/Preintuitionism ”This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano's axiomatic theory of natural numbers.
Peano's fifth axiom states:
* Allow that; zero has a property P;
* And; if every natural number less than a number x has the property P then x also has the property P.
* Therefore; every natural number has the property P.
This is the principle of complete induction, it establishes the property of induction as necessary to the system. Since Peano's axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong to any x and also x+1. What one can do is say that, if after some number n of trails that show a property P conserved in x and x+1, then we may infer that it will still hold to be true after n+1 trails. But this is itself induction. And hence the argument is a vicious circle.
From this Poincaré argues that if we fail to establish the consistency of Peano's axioms for natural numbers without falling into circularity, then the principle of complete induction is improvable by general logic. “
GODEL ACCEPTED IMPREDICATIVE DEFINITIONS
quote
http://www.friesian.com/goedel/chap-1.htm ”recent research [9] has shown that more can be squeezed out of these restrictions than had been expected:
all mathematically interesting statements about the natural numbers, as well as many analytic statements, which have been obtained by impredicative methods can already be obtained by predicative ones.[10]
We do not wish to quibble over the meaning of "mathematically interesting." However, "it is shown that the arithmetical statement expressing the consistency of predicative analysis is provable by impredicative means." Thus it can be proved conclusively that restricting mathematics to predicative methods does in fact eliminate a substantial portion of classical mathematics.[11]
Gödel has offered a rather complex analysis of the vicious circle principle and its devastating effects on classical mathematics culminating in the conclusion that because it "destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a good deal of modern mathematics itself" he would "consider this rather as a proof that the vicious circle principle is false than that classical mathematics is false."[12]”
Gödel is a complete failure as he ends in utter meaninglessness. His meaningless/paradoxical result comes directly from using axioms that lead or end in paradox. Even if Godel did not prove that mathematics was inconsistent Gödel proved nothing as it was totality built upon invalid axioms; All talk of what Godel achieved is just another myth mathematicians foist upon an ignorant population to beguile them into believing mathematician know what they are talking about and have access to truth.
GODEL IS SELF-CONTRADICTORY
But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done
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